We let \(\Sigma ^0_1\)  be the collection of open sets and \(\Pi ^0_1={\check{\Sigma }}^{\raisebox {-2pt}{\scriptsize {0}}}_1= \lbrace  X\setminus A\colon A \in \Sigma ^0_1\rbrace \)  be the collection of closed sets.
In general, for \(\alpha <\omega _1\)  we let \(\Sigma ^0_\alpha \)  be the collection of
countable unions \(A=\bigcup _n A_n\)  where each \(A_n \in \Pi ^0_{\alpha _n}\) 
for some \(\alpha _n<\alpha \) . We also let \(\Pi ^0_\alpha = {\check{\Sigma }}^{\raisebox {-2pt}{\scriptsize {0}}}_\alpha = \lbrace  X\setminus A\colon A \in \Sigma ^0_\alpha \rbrace \) .
Equivalently, \(A\in \Pi ^0_\alpha \)  if \(A=\bigcap _n A_n\)  where
\(A_n\in \Sigma ^0_{\alpha _n}\)  and each \(\alpha _n<\alpha \) .
We also set \(\Delta ^0_\alpha = \Sigma ^0_\alpha \cap \Pi ^0_\alpha \) . In particular,
\(\Delta ^0_1\)  is the collection of clopen sets.
In classical terminology, \(\Sigma ^0_2\)  is the collection of \(F_\sigma \)  sets and \(\Pi ^0_2\) 
is the collection of \(G_\delta \)  sets.
For any topological space, \(\mathcal {B}(X)=\bigcup _{\alpha <\omega _1} \Sigma ^0_\alpha =\bigcup _{\alpha <\omega _1}\Pi ^0_\alpha \) . A basic fact (see [1]})
is that for any uncountable Polish space \(X\) , all of the classes \(\Delta ^0_\alpha \) , \(\Sigma ^0_\alpha \) , \(\Pi ^0_\alpha \) , for \(\alpha <\omega _1\) ,
are distinct. We say a set
\(A\subseteq X\)  is \(\Sigma ^0_\alpha \)  (resp. \(\Pi ^0_\alpha \) ) hard
if \(A \notin \Pi ^0_\alpha \)  (resp. \(A\notin \Sigma ^0_\alpha \) ). This says that \(A\)  is
“no simpler” than a \(\Sigma ^0_\alpha \)  set. We say that \(A\)  is \(\Sigma ^0_\alpha \) -complete
if \(A\in \Sigma ^0_\alpha \setminus \Pi ^0_\alpha \) , that is, \(A \in \Sigma ^0_\alpha \)  and
\(A\)  is \(\Sigma ^0_\alpha \)  hard. This says that \(A\)  is exactly at the complexity level
\(\Sigma ^0_\alpha \) . Likewise, \(A\)  is \(\Pi ^0_\alpha \) -complete if \(A\in \Pi ^0_\alpha \setminus \Sigma ^0_\alpha \) .
